Finite Aperiodic Semigroups
A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups, so a synonym used (only?) in such contexts is group-free semigroup. In terms of Green's relations, a finite semigroup is aperiodic if and only if its H-relation is trivial. These two characterizations extend to group-bound semigroups.
A celebrated result of algebraic automata theory due to Marcel-Paul Schützenberger asserts that a language is star-free if and only if its syntactic monoid is finite and aperiodic.
A consequence of the Krohn–Rhodes theorem is that every finite aperiodic monoid divides a wreath product of copies of the three element monoid containing an identity element and two right zeros. The two-sided Krohn–Rhodes theorem alternatively characterizes finite aperiodic monoids as divisors of iterated block products of copies of the two-element semilattice.
Read more about this topic: Aperiodic Semigroup
Famous quotes containing the word finite:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)